# Schoenflies and symplectic topology

The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the Schönflies conjecture? Can we deform any $S^3$ to be pseudo-convex?

My question is simply about interpreting Hutchings' latter "deformation" question and its connection to the Schoenflies conjecture. Suppose $P \subset S^4$ is a smoothly embedded $S^3$, which we know bounds a 4-manifold $X$ homeomorphic to $B^4$ by a theorem of Brown. Viewing $P$ inside $\mathbb{C}^2 = S^4-\{pt\}$, is Hutchings asking whether we can perturb $P$ so that it is the pseudoconvex boundary of a Stein domain?

A positive answer would imply (the smooth 4-dimensional) Schoenflies conjecture, right? First, a Stein-fillable (or weakly/strongly symplectically fillable, for that matter) contact structure on $S^3$ must be tight (Eliashberg-Gromov), hence isotopic to $(S^3,\xi_{\operatorname{std}})$. Now since a Stein filling of $(S^3,\xi_{\operatorname{std}})$ must be diffeomorphic to $B^4$ (Eliashberg), we conclude that $P$ bounds a smoothly standard $B^4$.

Finally, this seems to suggest a (perhaps) weaker question: Can we perturb $P$ and endow it with a contact structure making $X$ into a weak symplectic filling? A positive answer again seems to imply Schoenflies. Indeed, the above arguments would imply that $P$ is a standard contact $S^3$. Note that $X$ is homeomorphic to $B^4$, so it is certainly symplectically aspherical for any smooth and symplectic structure. By Eliashberg (perhaps Eliashberg-McDuff?), a symplectically aspherical filling of $(S^3,\xi_{\operatorname{std}})$ is diffeomorphic to $B^4$.

• Generally speaking this isn't a place to post giant, long known open problems. Usually such questions are closed quite rapidly. Commented Mar 26, 2015 at 20:51
• @RyanBudney: Sorry if my post is unclear; I'll try to revise it. I am asking if my interpretation of Hutchings' question is correct and if an affirmative answer to it would imply Schoenflies. Commented Mar 26, 2015 at 20:53
• Oh, okay, sorry I originally thought your question was a request to fill in the details, i.e. actually prove the smooth Schoenflies hypothesis in dimension 4. I'm pretty sure your interpretation is correct. Commented Mar 26, 2015 at 20:57
• @RyanBudney: Thanks. Also, I'll remove the secondary question about whether or not an affirmative answer would imply (smooth 4D) Schoenflies if that's considered off-topic. Commented Mar 26, 2015 at 21:05
• Your interpretation of the first part is correct. It is really just a case of remark 4.8 here arxiv.org/abs/1508.01491. It should be said that there are no known examples of contractible compact 4 manifolds (regardless of the boundary) which do not admit Stein structures and as of now this seems to be a wide open question if any such examples exist.
– PVAL
Commented Sep 30, 2015 at 18:18